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/* diff/diff.c
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*
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* Copyright (C) 1996, 1997, 1998, 1999, 2000 David Morrison
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*
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* This program is free software; you can redistribute it and/or modify
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* it under the terms of the GNU General Public License as published by
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* the Free Software Foundation; either version 3 of the License, or (at
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* your option) any later version.
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*
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* This program is distributed in the hope that it will be useful, but
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* WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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* General Public License for more details.
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*
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* You should have received a copy of the GNU General Public License
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* along with this program; if not, write to the Free Software
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* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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*/
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#include <config.h>
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#include <stdlib.h>
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#include <gsl/gsl_math.h>
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#include <gsl/gsl_errno.h>
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#undef GSL_DISABLE_DEPRECATED
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#include <gsl/gsl_diff.h>
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int
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gsl_diff_backward (const gsl_function * f,
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double x, double *result, double *abserr)
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{
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/* Construct a divided difference table with a fairly large step
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size to get a very rough estimate of f''. Use this to estimate
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the step size which will minimize the error in calculating f'. */
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int i, k;
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double h = GSL_SQRT_DBL_EPSILON;
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double a[3], d[3], a2;
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/* Algorithm based on description on pg. 204 of Conte and de Boor
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(CdB) - coefficients of Newton form of polynomial of degree 2. */
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for (i = 0; i < 3; i++)
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{
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a[i] = x + (i - 2.0) * h;
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d[i] = GSL_FN_EVAL (f, a[i]);
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}
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for (k = 1; k < 4; k++)
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{
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for (i = 0; i < 3 - k; i++)
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{
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d[i] = (d[i + 1] - d[i]) / (a[i + k] - a[i]);
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}
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}
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/* Adapt procedure described on pg. 282 of CdB to find best value of
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step size. */
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a2 = fabs (d[0] + d[1] + d[2]);
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if (a2 < 100.0 * GSL_SQRT_DBL_EPSILON)
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{
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a2 = 100.0 * GSL_SQRT_DBL_EPSILON;
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}
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h = sqrt (GSL_SQRT_DBL_EPSILON / (2.0 * a2));
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if (h > 100.0 * GSL_SQRT_DBL_EPSILON)
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{
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h = 100.0 * GSL_SQRT_DBL_EPSILON;
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}
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*result = (GSL_FN_EVAL (f, x) - GSL_FN_EVAL (f, x - h)) / h;
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*abserr = fabs (10.0 * a2 * h);
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return GSL_SUCCESS;
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}
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int
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gsl_diff_forward (const gsl_function * f,
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double x, double *result, double *abserr)
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{
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/* Construct a divided difference table with a fairly large step
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size to get a very rough estimate of f''. Use this to estimate
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the step size which will minimize the error in calculating f'. */
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int i, k;
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double h = GSL_SQRT_DBL_EPSILON;
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double a[3], d[3], a2;
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/* Algorithm based on description on pg. 204 of Conte and de Boor
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(CdB) - coefficients of Newton form of polynomial of degree 2. */
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for (i = 0; i < 3; i++)
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{
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a[i] = x + i * h;
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d[i] = GSL_FN_EVAL (f, a[i]);
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}
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for (k = 1; k < 4; k++)
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{
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for (i = 0; i < 3 - k; i++)
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{
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d[i] = (d[i + 1] - d[i]) / (a[i + k] - a[i]);
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}
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}
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/* Adapt procedure described on pg. 282 of CdB to find best value of
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step size. */
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a2 = fabs (d[0] + d[1] + d[2]);
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if (a2 < 100.0 * GSL_SQRT_DBL_EPSILON)
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{
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a2 = 100.0 * GSL_SQRT_DBL_EPSILON;
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}
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h = sqrt (GSL_SQRT_DBL_EPSILON / (2.0 * a2));
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if (h > 100.0 * GSL_SQRT_DBL_EPSILON)
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{
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h = 100.0 * GSL_SQRT_DBL_EPSILON;
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}
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*result = (GSL_FN_EVAL (f, x + h) - GSL_FN_EVAL (f, x)) / h;
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*abserr = fabs (10.0 * a2 * h);
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return GSL_SUCCESS;
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}
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int
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gsl_diff_central (const gsl_function * f,
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double x, double *result, double *abserr)
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{
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/* Construct a divided difference table with a fairly large step
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size to get a very rough estimate of f'''. Use this to estimate
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the step size which will minimize the error in calculating f'. */
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int i, k;
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double h = GSL_SQRT_DBL_EPSILON;
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double a[4], d[4], a3;
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/* Algorithm based on description on pg. 204 of Conte and de Boor
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(CdB) - coefficients of Newton form of polynomial of degree 3. */
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for (i = 0; i < 4; i++)
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{
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a[i] = x + (i - 2.0) * h;
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d[i] = GSL_FN_EVAL (f, a[i]);
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}
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for (k = 1; k < 5; k++)
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{
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for (i = 0; i < 4 - k; i++)
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{
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d[i] = (d[i + 1] - d[i]) / (a[i + k] - a[i]);
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}
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}
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/* Adapt procedure described on pg. 282 of CdB to find best
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value of step size. */
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a3 = fabs (d[0] + d[1] + d[2] + d[3]);
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if (a3 < 100.0 * GSL_SQRT_DBL_EPSILON)
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{
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a3 = 100.0 * GSL_SQRT_DBL_EPSILON;
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}
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h = pow (GSL_SQRT_DBL_EPSILON / (2.0 * a3), 1.0 / 3.0);
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if (h > 100.0 * GSL_SQRT_DBL_EPSILON)
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{
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h = 100.0 * GSL_SQRT_DBL_EPSILON;
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}
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*result = (GSL_FN_EVAL (f, x + h) - GSL_FN_EVAL (f, x - h)) / (2.0 * h);
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*abserr = fabs (100.0 * a3 * h * h);
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return GSL_SUCCESS;
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}
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